scholarly journals Existence of least energy nodal solution with two nodal domains for a generalized Kirchhoff problem in an Orlicz-Sobolev space

2016 ◽  
Vol 290 (4) ◽  
pp. 583-603 ◽  
Author(s):  
Giovany M. Figueiredo ◽  
Jefferson A. Santos
Keyword(s):  
Sobolev Space ◽  
Nodal Solution ◽  
Nodal Domains ◽  
Least Energy ◽  
Kirchhoff Problem

scholarly journals Morse index of radial nodal solutions of Hénon type equations in dimension two

2017 ◽  
Vol 19 (03) ◽  
pp. 1650042 ◽  
Author(s):  
Ederson Moreira dos Santos ◽  
Filomena Pacella
Keyword(s):  
Elliptic Equations ◽  
Morse Index ◽  
Radial Solution ◽  
Index Formula ◽  
Connected Components ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Semilinear Elliptic ◽  
Bounded Sets ◽  
Least Energy

We consider non-autonomous semilinear elliptic equations of the type [Formula: see text] where [Formula: see text] is either a ball or an annulus centered at the origin, [Formula: see text] and [Formula: see text] is [Formula: see text] on bounded sets of [Formula: see text]. We address the question of estimating the Morse index [Formula: see text] of a sign changing radial solution [Formula: see text]. We prove that [Formula: see text] for every [Formula: see text] and that [Formula: see text] if [Formula: see text] is even. If [Formula: see text] is superlinear the previous estimates become [Formula: see text] and [Formula: see text], respectively, where [Formula: see text] denotes the number of nodal sets of [Formula: see text], i.e. of connected components of [Formula: see text]. Consequently, every least energy nodal solution [Formula: see text] is not radially symmetric and [Formula: see text] as [Formula: see text] along the sequence of even exponents [Formula: see text].

scholarly journals Nodal Solutions for Sublinear-Type Problems with Dirichlet Boundary Conditions

2020 ◽  
Author(s):  
Denis Bonheure ◽  
Ederson Moreira dos Santos ◽  
Enea Parini ◽  
Hugo Tavares ◽  
Tobias Weth
Keyword(s):  
Boundary Conditions ◽  
Morse Index ◽  
Symmetric Functions ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Nodal Solution ◽  
Axially Symmetric ◽  
Nodal Solutions ◽  
Dirichlet Eigenvalue ◽  
Least Energy

Abstract We consider nonlinear 2nd-order elliptic problems of the type $$\begin{align*} & -\Delta u=f(u)\ \textrm{in}\ \Omega, \qquad u=0\ \textrm{on}\ \partial \Omega, \end{align*}$$where $\Omega $ is an open $C^{1,1}$–domain in ${{\mathbb{R}}}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=|s|^{p-1}s$ with $0<p<1$ and of Allen–Cahn type $f(s)=\lambda (s-|s|^{p-1}s)$ with $p>1$ and $\lambda>\lambda _2(\Omega )$ (the second Dirichlet eigenvalue of the Laplacian). We prove the existence of a least energy nodal (i.e., sign changing) solution and of a nodal solution of mountain-pass type. We then give explicit examples of domains where the associated levels do not coincide. For the case where $\Omega $ is a ball or annulus and $f$ is of class $C^1$, we prove instead that the levels coincide and that least energy nodal solutions are nonradial but axially symmetric functions. Finally, we provide stronger results for the Allen–Cahn type nonlinearities in case $\Omega $ is either a ball or a square. In particular, we give a complete description of the solution set for $\lambda \sim \lambda _2(\Omega )$, computing the Morse index of the solutions.

Nodal Solutions for a Quasilinear Elliptic Equation Involving the p-Laplacian and Critical Exponents

2018 ◽  
Vol 18 (1) ◽  
pp. 17-40
Author(s):  
Yinbin Deng ◽  
Shuangjie Peng ◽  
Jixiu Wang
Keyword(s):  
Elliptic Equations ◽  
Order Term ◽  
Quasilinear Elliptic Equations ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Content Type ◽  
Nodal Domains ◽  
Change Of Variables ◽  
Graphic Content ◽  
Quasilinear Elliptic

AbstractThis paper is concerned with the following type of quasilinear elliptic equations in{\mathbb{R}^{N}}involving thep-Laplacian and critical growth:-\Delta_{p}u+V(|x|)|u|^{p-2}u-\Delta_{p}(|u|^{2})u=\lambda|u|^{q-2}u+|u|^{2p^{% *}-2}u,which arises as a model in mathematical physics, where{2<p<N},{p^{*}=\frac{Np}{N-p}}. For any given integer{k\geq 0}, by using change of variables and minimization arguments, we obtain, under some additional assumptions onpandq, a radial sign-changing nodal solution with{k+1}nodal domains. Since the critical exponent appears and the lower order term (obtained by a transformation) may change sign, we shall use delicate arguments.

scholarly journals The Existence of Least Energy Sign-Changing Solution for Kirchhoff-Type Problem with Potential Vanishing at Infinity

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Ting Xiao ◽  
Canlin Gan ◽  
Qiongfen Zhang
Keyword(s):  
Variational Method ◽  
Type Equation ◽  
Previous Literature ◽  
Type Problem ◽  
Nodal Domains ◽  
Kirchhoff Type ◽  
Deformation Lemma ◽  
Kirchhoff Type Problem ◽  
Quantitative Deformation Lemma ◽  
Least Energy

In this paper, we study the Kirchhoff-type equation: − a + b ∫ ℝ 3     ∇ u 2 d x Δ u + V x u = Q x f u , in   ℝ 3 , where a , b > 0 , f ∈ C 1 ℝ 3 , ℝ , and V , Q ∈ C 1 ℝ 3 , ℝ + . V x and Q x are vanishing at infinity. With the aid of the quantitative deformation lemma and constraint variational method, we prove the existence of a sign-changing solution u to the above equation. Moreover, we obtain that the sign-changing solution u has exactly two nodal domains. Our results can be seen as an improvement of the previous literature.

scholarly journals Least-energy nodal solutions of critical Kirchhoff problems with logarithmic nonlinearity

2020 ◽  
Vol 10 (4) ◽  
Author(s):  
Sihua Liang ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
State Energy ◽  
Convergence Property ◽  
Degree Theory ◽  
Critical Sobolev Exponent ◽  
Topological Degree Theory ◽  
Nodal Solutions ◽  
Sobolev Exponent ◽  
Sign Changing Solutions ◽  
Least Energy ◽  
Kirchhoff Problem

AbstractIn this paper, we are concerned with the existence of least energy sign-changing solutions for the following fractional Kirchhoff problem with logarithmic and critical nonlinearity: $$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_{s,p}^p\right) (-\Delta )^s_pu = \lambda |u|^{q-2}u\ln |u|^2 + |u|^{ p_s^{*}-2 }u &{}\quad \text {in } \Omega , \\ u=0 &{}\quad \text {in } {\mathbb {R}}^N{\setminus } \Omega , \end{array}\right. \end{aligned}$$ a + b [ u ] s , p p ( - Δ ) p s u = λ | u | q - 2 u ln | u | 2 + | u | p s ∗ - 2 u in Ω , u = 0 in R N \ Ω , where $$N >sp$$ N > s p with $$s \in (0, 1)$$ s ∈ ( 0 , 1 ) , $$p>1$$ p > 1 , and $$\begin{aligned}{}[u]_{s,p}^p =\iint _{{\mathbb {R}}^{2N}}\frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}dxdy, \end{aligned}$$ [ u ] s , p p = ∬ R 2 N | u ( x ) - u ( y ) | p | x - y | N + p s d x d y , $$p_s^*=Np/(N-ps)$$ p s ∗ = N p / ( N - p s ) is the fractional critical Sobolev exponent, $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N $$(N\ge 3)$$ ( N ≥ 3 ) is a bounded domain with Lipschitz boundary and $$\lambda $$ λ is a positive parameter. By using constraint variational methods, topological degree theory and quantitative deformation arguments, we prove that the above problem has one least energy sign-changing solution $$u_b$$ u b . Moreover, for any $$\lambda > 0$$ λ > 0 , we show that the energy of $$u_b$$ u b is strictly larger than two times the ground state energy. Finally, we consider b as a parameter and study the convergence property of the least energy sign-changing solution as $$b \rightarrow 0$$ b → 0 .

Existence of a Least Energy Nodal Solution for a Class of p&q-Quasilinear Elliptic Equations

2014 ◽  
Vol 14 (2) ◽  
Author(s):  
Sara Barile ◽  
Giovany M. Figueiredo
Keyword(s):  
Bounded Domain ◽  
Elliptic Equations ◽  
Existence Result ◽  
Quasilinear Elliptic Equations ◽  
Nodal Solution ◽  
Smooth Bounded Domain ◽  
Quasilinear Elliptic ◽  
Least Energy

AbstractIn this paper we prove an existence result for a least energy nodal (or sign-changing) solution for the class of p&q problems given bywhere Ω is a smooth bounded domain in ℝ

scholarly journals Discontinuous perturbations of nonhomogeneous strongly-singular Kirchhoff problems

2021 ◽  
Vol 28 (6) ◽  
Author(s):  
Vicenţiu D. Rădulescu ◽  
Carlos Alberto Santos ◽  
Lais Santos ◽  
Marcos L. M. Carvalho
Keyword(s):  
Sobolev Space ◽  
Convex Analysis ◽  
Singular Term ◽  
Finite Energy ◽  
Energy Functional ◽  
Singular Problem ◽  
Discontinuous Nonlinearity ◽  
Generalized Gradient ◽  
Whole Space ◽  
Kirchhoff Problem

AbstractIn this paper, we are concerned with a Kirchhoff problem in the presence of a strongly-singular term perturbed by a discontinuous nonlinearity of the Heaviside type in the setting of Orlicz–Sobolev space. The presence of both strongly-singular and non-continuous terms brings up difficulties in associating a differentiable functional to the problem with finite energy in the whole space $$W_0^{1,\Phi }(\Omega )$$ W 0 1 , Φ ( Ω ) . To overcome this obstacle, we establish an optimal condition for the existence of $$W_0^{1,\Phi }(\Omega )$$ W 0 1 , Φ ( Ω ) -solutions to a strongly-singular problem, which allows us to constrain the energy functional to a subset of $$W_0^{1,\Phi }(\Omega )$$ W 0 1 , Φ ( Ω ) in order to apply techniques of convex analysis and generalized gradient in the sense of Clarke.

Least Energy Nodal Solutions for a Defocusing Schrödinger Equation with Supercritical Exponent

2018 ◽  
Vol 62 (1) ◽  
pp. 1-23 ◽  
Author(s):  
Minbo Yang ◽  
Carlos Alberto Santos ◽  
Jiazheng Zhou
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Positive Function ◽  
Nehari Manifold ◽  
Nodal Solution ◽  
Nodal Solutions ◽  
Quasilinear Schrödinger Equation ◽  
Subcritical Growth ◽  
Image Position ◽  
Least Energy

AbstractIn this paper we consider the existence of least energy nodal solution for the defocusing quasilinear Schrödinger equation $$-\Delta u - u \Delta u^2 + V(x)u = a(x)[g(u) + \lambda \vert u \vert ^{p-2}u] \hbox{in} {\open R}^N,$$ where λ≥0 is a real parameter, V(x) is a non-vanishing function, a(x) can be a vanishing positive function at infinity, the nonlinearity g(u) is of subcritical growth, the exponent p≥22*, and N≥3. The proof is based on a dual argument on Nehari manifold by employing a deformation argument and an $L</italic>^{\infty}({\open R}^{N})$-estimative.

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